Performance oriented anti-windup for a class of neural network controlled systems

Performance oriented anti-windup for a Performance oriented anti-windup for a class of neural network controlled systemsclass of neural network controlled systems

G. HerrmannM. C. Turner and I. Postlethwaite

Control and Instrumentation Research GroupUniversity of Leicester

SWAN 2006SWAN 2006 -

Automation and Robotics Research Institute, UTA

Anti-windup for a class of neural network controlled systems2

1. Motivation

2. The plant: A linear plant with matched unknown non-linearities

3. The nominal control system: Linear Control with augmented NN-controller for disturbance rejection

4. Controller conditioning for anti-windup:

– Preliminaries: Constrained multi-variable systems

– Non-linear Controller Conditioning

– Linear Controller Conditioning

5. An Example

6. Conclusions


Unknown Nonlinearity

++

MotivationMotivation

LinearPlant

LinearController +

NNcompen-

sation

-

Adap-tation

NN-Control- Examples :S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. World Scientific, Singapore, 1998.

Y. Kim and F.L. Lewis, High-Level Feedback Control with Neural Networks," World Scientific, Singapore, 1998.

??Cu u

Linear control performance in combination with NN-control – Examples of practical validation:G. Herrmann, S. S. Ge, and G. Guo, “Practical implementation of a neural network controller in a hard disk drive,” IEEE Transactions on Control Systems Technology, 2005. ——, “A neural network controller augmented to a high performance linear controller and its application to a HDD-track following servo system,” IFAC 2005 (under journal review).

Anti-Windup (AW)(AW) Control - a possible approach to overcome controller saturationG. Grimm, J. Hatfield, I. Postlethwaite, A. R. Teel, M. C. Turner, and L. Zaccarian, “Antiwindup for stable linear systems with input saturation: An LMI based synthesis,” IEEE Trans. on Autom. Control, vol. 48, no. 9, pp. 1509–1525, 2003.Alternative for NN:W. Gao; R.R. Selmic, "Neural network control of a class of nonlinear systems with actuator saturation Neural Networks", IEEE Trans. on Neural Networks, Vol. 17, No. 1, 2006.


LinearPlant

LinearController

+ -

LinearAW-Compen-

sator

Cu u

Motivation: Principle of anti-windup compensationMotivation: Principle of anti-windup compensation


The plantThe plant

,

,

);(

pyu

Pp

ppPpP

nnn

xCy

yfBuBxAx

Stable, minimum-phase, strictly proper with matched nonlinear disturbance f(y)

upp nBCrank )(


The plantThe plant

*W - optimal (constant) weight matrix

)(s - neural network basis function vector,

- neural network modelling error

so that it can be arbitrarily closely modelled by a neural network approach:

;)()( * ysWyfT ;lssT fKK

The disturbance is continuous in y and bounded:

fKyf )(


The Nominal Controller – Linear Control ComponentThe Nominal Controller – Linear Control Component

,

;

,

,

dDyDxCu

dByBxAx

dCCCCL

dCCCCC

PCPPCP

PCCCL CDBACB

CBAA is assumed to be Hurwitz stable

d - exogenous demand signal

The linear controller component defines the closed loop steady state:

dDB

BACy

dCP

dCCLPd

,

,10 dDB

BA

x

x

dCP

dCCL

dP

dC

,

,1

,

,

dPP

dCC

P

C

xx

xx

x

x

,

,

~

~and the controller error:


The Nominal Controller – Non-Linear Control ComponentThe Nominal Controller – Non-Linear Control Component

LNL uuu

Estimation algorithm: )(0000ˆdii yyNs(y)ΓW

lliΓ R is symmetric, positive definite Learning Coefficient Matrix

, ]ˆˆˆ[ˆ21 unWWWW ][ 31

unWWWW

unWWWWW

~~~ˆ21 - Estimation error

estimate -compensatesfor non-linearity

discontinuous sliding mode component - compensates for modeling error

NMatrix is a design parameter

)(

)( )(ˆ

d

dTNL yyN

yyNKysWu


The Nominal ControllerThe Nominal Controller

dPP

dCC

P

C

xx

xx

x

x

,

,

~

~

can asymptotically track the signal yd so that the controller error:

becomes zero.

unWWWWW

~~~ˆ21

The estimation error

remains bounded.


+

NNcompen-

sation

-


++

Controller conditioningController conditioning

LinearPlant

LinearController

Adap-tation

Cu u

Non-linear

Algorithm

Linear AW-comp.

+ -


Controller conditioning - PreliminariesController conditioning - Preliminaries

Multi-variable Saturation Function:

)(

)(

)(

,

1,

,

11

uununnuu

uu

uu

usat

usat

uSat )),,max(min()(

, iiiiuuuuuusat

ii

iu

iu

iu

iuSymmetric Multi-variable Saturation Function:

)()( :,

uSatuSatuuuuu

The Deadzone - Counter-part of a Saturation Function:

)()(

);()(,,

uSatuuDZ

uSatuuDZ

uu

uuuu


Controller conditioning - Controller conditioning - AssumptionsAssumptions

+

NNcompen-

sation

-


++

LinearPlant

LinearController

Adap-tation

Saturation Limit:Saturation Limit: Cu u

);( CuuSatu

unu

u

u 1

Disturbance LimitDisturbance Limit

)(min,...,1

ini

f uKu

The controller amplitude is large enough to compensate for the unknown non-linearity.

Permissible Range of Tracking Control SystemPermissible Range of Tracking Control System

dDdDB

BACDCu dC

dCP

dCCLPCCdL ,

,

,1,

We do not assume that

the transient behaviour has to satisfy this constraint.0 small design parameter );(0 ,)1( dLKu

uDZf


Controller conditioning – Controller conditioning – Non-linear Control ElementNon-linear Control Element

)(

)(||)~(||1-

))~())(~((~

||)~(||

2

d

df

NLKfNLKf

fNLNL yyN

yyNK

NLuc

uDzKuDzK

Kuu

NLuc

ff

)(

)()(ˆ~

d

dTNL yyN

yyNKysWu

iidLKuidii WΓuDZyyNs(y)ΓNL

ucWf

ˆ)()(0000||)~(||ˆ,)1(

0 is a small design dependent constant

0||)~(|| ~ , ~ NL

ucuKu NLfNL

and replaced by a high gain controller. The NN-estimation algorithm is slowed down.The NN-controller is cautiously disabled

0)~( NLK uDzf

1||)~(|| ;~ NL

ucuu NLNL NN-control is usedNN-control is used

)1( fNL Ku


Controller conditioning – Controller conditioning – Linear Control ElementLinear Control Element

Linear controller

;)(

))((42

41

32

31

21

2

1

Luuuu

AA

uDZ

x

v

vx

NLNL

AW-compensator:

in practice 0

Note that ,0))(())((

NLLuuuuuuuSatDZ

NLNLThe control limits are satisfied

to be designed

,

,

dDyDxCu

dByBxAx

dCCCCCL

dCCCCCC

Closed Loop: ,)( ; ))(( NLLuuuuC uwuSatuyy

NLNL

compensation

,

;

2,

1,

wvdDyDxCu

vdByBxAx

dCCCCCL

dCCCCCC

withcompensation signals


Controller conditioning – Controller conditioning – AW-Compensator Design TargetAW-Compensator Design Target

+

NNcompen-

sation

-


++

LinearPlant

LinearController

Adap-tation

Linear AW-comp. + -

Cu u

Non-linear

Algorithm

++ -

wz

dy

Linear AW-comp.

where PzCz xCxCz ~ ~21 is a designer chosen performance output

Design target for linear Design target for linear AW-compensator:AW-compensator:

0, ;)()( 2

0

22

0

2

dsswdsszMinimize for

This L2-gain optimization target ensures recovery of the nominal controller performance.


Controller conditioning – Controller conditioning – AW-Compensator Design TargetAW-Compensator Design Target

The conditioned linear control uL term operating in connection with the constrained

NN-controller uNL, will track asymptotically any permissible steady state.

The NN-weight estimates will remain bounded.

Design target for overall AW-compensator:Design target for overall AW-compensator:

+

NNcompen-

sation

-


++

LinearPlant

LinearController

Adap-tation

Linear AW-comp. + -

Cu u

Non-linear

Algorithm

++ -d

y


A Simulation ExampleA Simulation Example

Simulation for a direct drive DC-torque motor[12] Hsieh & Pan (2000)

Hsieh & Pan (2000) [12]:

6-th order model to include issues of static friction, i.e. the pre-sliding behaviour:

650;=k

0.175;=C

;015.54=m

1

s

-4

The nominal model used for linear controller design

;1010

2

11

2

1 umx

x

m

C

m

kx

xs

Other parameters:

80000;=

50000;=k

2.5;=

454.5;=

4;=n

2

Assume both angle position x1 and angle velocity x2 are measurable



10000

1041943101 57

CA

00

0134217728CB

32

0,dCB

107 101250.31045.7 CC 0101 9CD 0, dCD

Nominal linear Controller:

Nominal NN-Controller:

2

222

2

22

1

2

1

ˆ)ˆ()(

exp

cxcxx

xsi

Gaussian RadialBasis Function

0005.0

001.0

2422191714129742

25212016151110651

cccccccccc

cccccccccc

;3.0ˆ ;15.0ˆˆ

;0ˆ ;15.0ˆ ;3.0ˆ

25212016

151110651

cccc

cccccc

;1.0ˆ;3/001.0



Saturation limit:5.1u

Conditioning of NN-Controller:

0005.0 ;1.0 ;3.1 KK f

1~xz

Linear AW-Compensator design:


A Simulation ExampleA Simulation ExampleControl signalPosition signal

0 0.02 0.04 0.06

-2

0

2

time

u

Unconstrained Response

0 0.02 0.04 0.06-3

-2

-1

0

1

2

3

u

time

Constrained Response

0 0.01 0.02 0.03 0.04 0.05 0.06-2

-1

0x 10

-4

x 1

time



0 0.02 0.04 0.06-1

-0.8

-0.6

-0.4

-0.2

0x 10

-3

x 1

time

Control signalPosition signal

0 0.02 0.04 0.06-15

-10

-5

0

5

10

15

time

u

Unconstrained Response

0 0.02 0.04 0.06-15

-10

-5

0

5

10

15

u

time

Constrained Response


ConclusionsConclusions

I. Development of a conditioning method for a linear controller & robust NN-controller combination:1. Nominal NN-controller: Add-on to a linear controller for compensation of

matched unknown non-linearities/disturbances

2. Linear controller conditioning: Specially structured AW-controller (considering former results)

3. NN-controller conditioning: The unknown non-linearity is bounded and can be counteracted by a variable structure component; once the NN-controller exceeds the bound.

II. Design target: 1. Retain asymptotic tracking for permissible demands and keep NN-estimates

bounded

2. Optimization of linear AW-controller according to an L2-constraint

III. Simulation Result: Performance similar for un/conditioned controller

Documents

Performance oriented anti-windup for a class of neural network controlled systems